On the rectilinear crossing number of complete graphs

Authors:
Uli Wagner
Affiliations:
Institut fü Theoretische Informatik, ETH Zürich, CH-8092 Zürich, Switzerland
Venue:
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2003

Citing 3
Cited 3

On the crossing number of complete graphs

Proceedings of the eighteenth annual symposium on Computational geometry
Lectures on Discrete Geometry

Lectures on Discrete Geometry
Toward the rectilinear crossing number of Kn: new drawings, upper bounds, and asymptotics

Discrete Mathematics

Note: Geometric drawings of Kn with few crossings

Journal of Combinatorial Theory Series A
The complexity of several realizability problems for abstract topological graphs

GD'07 Proceedings of the 15th international conference on Graph drawing
Improved bounds for the number of (≤ k)-sets, convex quadrilaterals, and the rectilinear crossing number of kn

GD'04 Proceedings of the 12th international conference on Graph Drawing

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Abstract

We prove a lower bound of 0.3288(n4) for the rectilinear crossing number cr(Kn) of a complete graph on n vertices, or in other words, for the minimum number of convex quadrilaterals in any set of n points in general position in the Euclidean plane. As we see it, the main contribution of this paper is not so much the concrete numerical improvement over earlier bounds, as the novel method of proof, which is not based on bounding cr(Kn) for some small n.